An \(s\) -subdivision of an edge replaces it by a path of length \(s+1\) .
In geometry and in particular in orthogonal [1]} and
RAC drawings [2]}, a subdivision is called a bend.
Each edge is replaced by a path of length at most \(s+1\)  in an
\(s\) -subdivision \(\sigma _s(G)\)  of graph \(G\) , where \(s\)  is the number of edge
for a subdivison of \(G\) . An \(s\) -subdivision is
uniform if each edge of graph \(G\) 
is extended to a path of length \(s+1\) . Then \(\sigma _s(G)\)  is a
single graph. In the general version, \(\sigma _s(G)\)  is parameterized
by its set of edges and is a set of graphs. Here these cases do not
matter, since too long paths can be folded, so that some of its
edges remain uncrossed. Subdivisions of graphs are used in
Kuratowski's theorem, which states that a planar graph does not
contain a subgraph that is a subdivision of \(K_5\)  or
\(K_{3,3}\) , i.e., \(K_5\)  and \(K_{3,3}\)  are the topological minors of
the planar graphs. Note that edge contractions, i.e., paths of
length zero, are generally used in the theory of graph minors
[3]}.
